Proof of Theorem REF .
Let \(\rho _{\rm glob}(k,n)\)  be as defined in (REF ),
and let \(\overline{\rho }_{\rm glob}(k,n)\)  be the same quantity with
the limit replaced by lim sup.
We write \(\rho _p(k,n)\)  for the probabilities computed in
Theorem REF . A standard argument (see for example
[1]}) uses the local conditions at finitely many primes to show that
\(\overline{\rho }_{\rm glob}(k,n) \leqslant \prod _{p < M} \rho _p(k,n).\) 
